Chromatin is of fundamental importance for a host of biological processes ranging from gene expression to cell division. Consequently there is huge research activity among biologists in this area. For physicists chromatin has also become of interest since there are more and more experiments available that work under quite well-defined conditions. Such experiments typically involve only a few components (DNA, histone proteins, etc) but no active protein
‘machines’. These experiments either focus on elucidating properties of single nucleosomes or of beads-on-a-string complexes (‘chromatin fibres’). They study the behaviour of these systems under changing ionic conditions and/or under an externally applied tension. Also the dynamics of these systems that is solely driven by thermal fluctuations is investigated.
Theoretical treatments and computer simulations that capture the essential features of the chromatin system are now possible and thus allow us to estimate the energy and timescales occurring in chromatin. Other approaches look at simplified model systems and try to identify general physical principles that govern complexes of charged chains and macro-ions. With the better understanding of the mechanical and dynamical properties of nucleosomes and chromatin fibres one hopefully also gains a deeper insight into more complicated questions like the working of chromatin remodelling complexes, the interaction between RNA polymerase and nucleosomes, etc.
To proceed in this direction it is crucial to obtain reliable numbers from experiments. One energy scale that dominates many processes is the adsorption energy of DNA on the octamer that has now been measured quite directly through stretching experiments. Another important feature, especially in 30 nm fibres, is the nucleosome–nucleosome interaction energy. Again detailed experimental studies have been performed and await a detailed theoretical treatment.
Chromatin is a very active and exciting field in biology where tremendous progress has been made in recent years. I hope that at least a few of the ideas gained from the physical models will be of help to biologists to develop a clearer picture of the working of chromatin and to design appropriate experimental setups.
Acknowledgments
The author thanks R Bruinsma, W M Gelbart, I Kuli´c and J Widom for many valuable discussions. Useful conversations with R Everaers, K Kremer, J Langowski, F Livolant, S Mangenot, G Maret and B Mergell are also acknowledged.
Appendix A. Rosette in d dimensions
The heterogeneity found for open loops (cf equation (44)) is reminiscent of a phase coexistence.
To clarify why the loop sizes are so sensitive to small details, especially why there is no ‘phase separation’ for closed chains (cf equation (41); even though d2f/dl2< 0 at larger separations), I will present here some unpublished results in which the free energy is formally recomputed for arbitrary space dimensions d. In order to do so one has to replace the entropy term in equation (38) by(d/2) ln(l/lP). Also in that case one obtains an analytical expression for the partition function, namely (for an open chain)
ZM= with Kν(x) denoting the modified Bessel function of νth order. In the case of large
‘pressure’ P, lPP/kBT 1, it follows from the asymptotic form of Kd/2−1(x) for large x , Kd/2−1(x) √
π/2xe−x, that the leading term of the resulting free energy G(P) is independent of d. Therefore one recovers the 3D case, i.e., equations (43)–(45) for L/M MlP. For the low ‘pressure’ regime, lPP/kBT 1, I use the asymptotics Kd/2−1(x) 2−1(|d/2 − 1|)(2/x)|d/2−1| for x 1 and d = 2. This leads to the following asymptotic behaviour of the partition function
ZM
It follows then from L= ∂G/∂ P that P is given in leading order by
P
The average leaf size can in principle be calculated, as before, from Z1. Here, however, it turns out to be more convenient to calculatellea f directly:
llea f = Now using the above given power law behaviour of the Bessel function together with equation (A.3) the average leaf size follows:
llea f
First note that for d < 2 the leaf size is set by the overall length of the chain but does not depend on lP; on the other hand, for d > 4llea f it is solely determined by lP. Speaking in the
picture of interacting particles on a track of length L one can explain these two extreme cases as follows. For d< 2 the increase of the ‘nearest neighbour pair potential’ beyond a distance lP (given by(d/2) ln(l/lP)) is too small to keep the particles together; instead they explore all available space. For higher space dimensions than 4 the prefactor of the log-term is large enough to keep neighbouring particles close to the ideal distance∼lP given by the shallow minimum of f(l), equation (38). The case d = 3, which I have already given in equation (44) (a result recovered in equation (A.5)), is an intermediate case wherellea f reflects the overall chain length L as well as the position∼lP of the shallow minimum. Note further that in the limit L→ ∞ the average size per leaf goes to infinity for d < 4 (but the ‘particles’ will only be spread out over the whole volume, Mllea f ≈ L, for d < 2).
Concerning the role of dimensionality one also gains some insight by the following simple argument (similar to the famous Onsager–Manning argument for the condensation of counterions on an infinitely long charged rod [111]). Consider a pair of particles at distance l in one dimension that attract each other via f(l) = kBT(d/2) ln(l/lP). Now assume that the particles move further apart from the distance l1to the distance l2 > l1. This leads to an increase in energy byE = kBT(d/2) ln(l2/l1). On the other hand the particles gain entropy since they are now less confined: −kBTS = kBT ln(l1/l2). Hence for d < 2 the particles will ‘lose’ each other since their attraction to the nearest neighbours is overruled by the gain in entropy, as derived rigorously in equation (A.5).
Finally, I mention that the same extension to arbitrary dimensions d can be performed for closed chains. One finds then phase separation for molten rosettes if d> 4. More specifically, loops have a preferred spacing L/N for d < 4 and 4χlP/(d − 4) for d > 4. This is different from the results on open chains, equation (A.5), in the interval 2< d < 4; hence in d = 3 molten rosettes respond strongly to a cutting of the chain.
Appendix B. Formation energy for small intranucleosomal loops
Since the configurations of small loops are essentially planar, it is convenient to describe them in terms of the function r(θ), cf figure 12(b), where r and θ are the polar coordinates of an arbitrary point on the loop (with the origin chosen on the cylinder axis and the X -axis running through the centre of the loop). In these terms the line element ds takes the form ds= dθ with primes and double primes denoting the first and second derivatives, respectively, with respect toθ. Restricting ourselves here to small loops, one can write r(θ) = R0 + u(θ) with u R0everywhere. Keeping only quadratic terms in u and its derivatives one obtains ds R0dθ(1 + u/R0+ u 2/(2R02)) and
The bending energy of the loop is then Eelastic 1
The variational energy of the loop F = Eelastic− T L, for a given aperture angle θ∗ and subject to the constraint of a fixed loop contour length L = L∗ +L, follows from equations (B.1) and (B.3) to be:
F{u(θ)} = 1 Here T is the Lagrange multiplier that constrains an extra lengthL to be adsorbed as the loop is formed (T can be interpreted as a ‘tension’ pulling in extra length). In equation (B.4) the constant terms as well as the term linear in u have been dropped since solutions to F with and without the linear u term differ just by a constant. The optimal loop shape (with given values ofL and θ∗) obeys the Euler–Lagrange equationδF/δu = 0:
u +
For vanishing ‘tension’, T = 0, one finds from equation (B.7) that the condition (B.8) is of the form 2 tan(√
2θ∗) = tan(θ∗/√
2) that has no non-trivial solution. For large T , T A/R02, one obtains from equations (B.7) and (B.8):
1
The left-hand side of equation (B.9) is small and hence solutions are approximately given by
T R20/Aθ∗ λ+θ∗ kπ with k = 1, 2, 3, . . .. In the following I consider only the k = 1 solution which is the solution that leads to the smallest elastic energy.
Using partial integration and equation (B.5) the loop formation energy, equation (60), can be cast into the form
To proceed further one makes use of the explicit solution given below equation (B.7). Assume the large tension case T A/R02 (the quality of this approximation will be checked a posteriori); thenλ− 1. The condition u(θ∗) = 0 takes then the form C2 −C1cos(λ+θ∗)
which leads, together withλ+θ∗ π, to C2 = C1. The loop shape is thus approximately given
Now minimizingU with respect to θ∗(forL fixed) gives the optimal aperture angle θ∗
Combining equations (B.12) and (B.13) one arrives at the final expression for the formation energy of a (small) loop of excess lengthL, equation (61).
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